Given $ m \angle RPS = 2x + 103$, and $ m \angle QPR = 9x - 11$, find $m\angle RPS$. $P$ $Q$ $S$ $R$
Answer: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Since $\angle QPS$ is a straight angle, we know ${m\angle QPS = 180}$ Substitute in the expressions that were given for each measure: $ {9x - 11} + {2x + 103} = {180}$ Combine like terms: $ 11x + 92 = 180$ Subtract $92$ from both sides: $ 11x = 88$ Divide both sides by $11$ to find $x$ $ x = 8$ Substitute $8$ for $x$ in the expression that was given for $m\angle RPS$ $ m\angle RPS = 2({8}) + 103$ Simplify: $ {m\angle RPS = 16 + 103}$ So ${m\angle RPS = 119}$.